Post a New Question

math

posted by on .

The point (x,y) lies on both conics x2+xy+x=81 and y2+xy+y=51. Given that x+y is positive, determine the value of x+y.

  • math - ,

    x ^ 2 + x y + x = 81

    x ( x + y + 1 ) = 81 Divide both sides by x

    x + y + 1 = 81 / x


    y ^ 2 + x y + y = 51

    y ( y + x + 1 ) = 51

    y ( x + y + 1 ) = 51 Divide both sides by y

    x + y + 1 = 51 / y


    x + y + 1 = x + y + 1


    81 / x = 51 / y Multiply both sides by x y

    81 x y / x = 51 x y / y

    81 y = 51 x Divide both sides by 81

    81 y / 81 = 51 x / 81

    y = 51 x / 81

    y = 3 * 17 * x / ( 3 * 27 )

    y = 17 x / 27

    Now put this value in formula :

    x ( x + y + 1 ) = 81

    x ( x + 17 x / 27 + 1 ) = 81

    x ( 27 x / 27 + 17 x / 27 + 1 ) = 81

    x ( 44 x / 27 + 1 ) = 81

    44 x ^ 2 / 27 + x = 81

    44 x ^ 2 / 27 + x - 81 = 0 Multiply both sides by 27

    44 x ^ 2 + 27 x - 2187 = 0

    The exact solutions of this equation are :

    x = - 81 / 11 and x = 27 / 4


    For x = - 81 / 11

    y = 17 x / 27 = - 51 / 11

    x + y = - 81 / 11 - 51 / 11 = - 132 / 11 = - 12


    For x = 27 / 4

    y = 17 x / 27 = 17 / 4

    x + y = 27 / 4 - 17 / 4 = 44 / 4 = 11


    x + y is positive so :

    x = 27 / 4 , y = 17 / 4

    x + y = 11

  • math - ,

    For x = 27 / 4

    y = 17 x / 27 = 17 / 4

    x + y = 27 / 4 + 17 / 4 = 44 / 4 = 11

Answer This Question

First Name:
School Subject:
Answer:

Related Questions

More Related Questions

Post a New Question